Financial Economics
Midterm Examination
1. (25) Suppose that you are a 17 year old student deciding whether or not to attend university a year from now. You may work from ages 18 to 65, and without a college degree you can earn 30, 000 per year. Assume that there are no taxes and that the interest rate is r = 0.05.
(a) (5) Calculate the present value of your lifetime earnings if you do not attend university.
(b) (5) Suppose that your university tuition would cost 50, 000 per year, and that you cannot work while in school, but that with a degree you can earn 50, 000 per year after graduation. What is the payback period on an investment in a college education, and is it worthwhile by the payback rule?
(c) (5) What is the net present value of going to college, and is it worthwhile by this metric?
(d) (5) Suppose now that the benefits of a college education are uncertain, and that upon graduation, rather than being guaranteed $50,000 per year, you will either find a job that pays $60,000 with probability p or one that pays $40,000 per year with probability 1 − p. Assuming that you are risk neutral, how high must p be for you to attend college?
(e) (5) Now suppose that you are risk averse, with a utility function u(W) = 1 − e
−100,000/W. How high must the probability p of finding a high paying job be for you to attend college?
2. (30) Armando is buying a house worth 500, 000, and must decide whether to take on a fixed or adjustable rate mortgage (ARM). The term of the loan is T = 15 years, with n = 12 monthly payments per year.
(a) (5) Let the fixed interest rate be r = 0.04. Calculate Armando’s monthly payment.
(b) (5) Suppose that the ARM has a starter rate of r0 = 0.03, guaranteed for at least the first 5 years. Calculate Armando’s initial monthly payment.
(c) (5) Now suppose that after 5 years, the rate jumps to r1 = 0.06. Calculate Armando’s new monthly payment, making sure to account for the payments that he has already made.
(d) (5) Now suppose that after 10 years, the rate jumps again to r2 = 0.09. Note that this would make the loan rather generous by the standards of ARMs, whose rates typically adjust as frequently as every month. Calculate Armando’s monthly payment now.
(e) (5) To evaluate whether he made the right decision, Armando considers discounting payments he made under the ARM using the fixed mortgage rate of r = 0.04. By this metric, was it a good idea to choose the ARM?
(f) (5) Assessing that the rate adjustments reflected fluctuations in the market rate over the life of the loan, Armando now considers discounting the payments he would have made under the fixed rate mortgage using the ARM rates. By this metric, was the ARM the right choice?
3. (25) Consider a portfolio choice problem with a risk-free asset with return rf and two risky assets, the first with mean return µ1 = 0.12 and standard deviation σ1 = 0.4 and the second with mean µ2 = 0.08 and standard deviation σ2 = 0.3, with correlation ρ12 = 0. For any stock portfolio let λ denote the proportion invested in stock 1.
(a) (5) Find the weight λ˜ that minimizes portfolio standard deviation σp.
(b) (5) Consider the tangency portfolio and let λ* denote the weight it places on stock 1. Find the condition that defines this value, but do not solve for it, and explain how it would compare to λ˜.
(c) (5) Now consider varying the risk-free rate rf . Again without solving anything, explain how you would expect λ* to vary as rf increases.
(d) (5) Show how the slope of the tangent line changes with rf . Recall a useful theorem that allows you to do this without ever actually solving for λ*.
(e) (5) Suppose instead that ρ12 = −1 so that the stocks always move against each other. Find the weight λf that yields a risk-free portfolio and the expected return µf to this portfolio.
4. (20) Suppose that Paul is a consumer solving a two period optimization problem, and that there are two potential states of the world next period, representing high H and low L consumption, that occur with probabilities πH = 0.4 and πL = 0.6. Suppose that there are two stocks, each with zero dividend, whose prices at present and in each future state are given below.
|
Price
|
Activision
|
Bethesda
|
|
Current
|
40
|
20
|
|
State H
|
10
|
50
|
|
State L
|
50
|
10
|
(a) (5) Using the law of one price, find what Paul’s stochastic discount factor M1 in states L and H must be.
(b) (5) Assume that Paul has log utility u(C) = log C and that his rate of time preference was δ = 0.9. If his consumption today is C0 = 1000, use the stochastic discount factor to calculate his future consumption C1 in states L and H.
(c) (5) Suppose that Paul receives no endowment next period, so that all of his future income comes from the payoffs to his investments. How many shares x of each stock must he hold for his consumption levels in each state to be what you found?
(d) (5) Calculate what Paul’s initial wealth A0 must be.